Bunches of Cones in the Divisor Class Group : a new Combinatorial Language for Toric Varieties

Abstract

As an alternative to the description of a toric variety by a fan in the lattice of one-parameter subgroups, we present a new language in terms of what we call bunches—these are certain collections of cones in the vector space of rational divisor classes. The correspondence between these bunches and fans is based on the classical Gale duality. The new combinatorial language allows a much more natural description of geometric phenomena around divisors of toric varieties than the usual method by fans does. For example, the numerically effective cone and the ample cone of a toric variety can be read off immediately from its bunch. Moreover, the language of bunches appears to be useful for classification problems.